The Fenchel duality theorem in Fréchet spaees

Abstract

We generalize the Fenchel duality and inf-convolution theorems from convex optimization to Fréchet spaces. Our approach is simple and direct, and is to first prove a more general theorem in which we consider only one function and replace the product of spaces by an arbitrary Fréchet space. These results extend those obtained recently by Attouch and Brezis. Applications of our Generalizations are given.

Keywords:

• Locally convex space
• Fréchet space
• fully barreled
• Legendbe-Fenchel transform
• inf-convolution
• effective domain
• proper function
• sublevel set
• slater con-straint condition
• subdifferential
• normal cone
• tangent cone
• Kuhn-Tucker condition

AMS 1980 Subject Classifications:

• Primary: 49 A 55
• Primary: 49 A 27
• Secondary: 90 C 25